Chain of Circles on a Chord

Here's Problem 1.3.3 from [Fukagawa & Pedoe]. The tablet, written in 1871, survives in the Ibaragi prefecture.

Points O1, O2, and O3 are collinear points and the circles O1(r), O2(r), and O3(r) touch each other, the first touching the second and the second the third. The circle O(R) circumscribes the three given circles, touching the first and the third internally. The chord PQ of this circle is an internal common tangent to the circles O1(r) and O3(r). Show that PQ = R + 3r.

Points O1, O2, and O3 are collinear points and the circles O1(r), O2(r), and O3(r) touch each other, the first touching the second and the second the third. The circle O(R) circumscribes the three given circles, touching the first and the third internally. The chord PQ of this circle is an internal common tangent to the circles O1(r) and O3(r). Show that PQ = R + 3r.

We shall generalize the problem a little. Instead of 3 circles we shall consider a chain of N touching circles, with N ≥ 3. This does not make the problem much more difficult.

So N circles of radius r are located on a chord, with midpoint O'; circle O(R) touches circles O1(r) and ON(r) internally; chord PQ is internally tangent to O1(r) and ON(r). Let W be the point of tangency of O(R) and ON(r), S be the point of tangency of PQ and ON(r). Let d denote the distance between O and O' and α be the angle QO'ON.

Naturally, R > d which forces the sign "+" before the radicals. We can now find PQ = s + t:

(2)

PQ = 2√R² - [d·cos(α)]²

Luckily, this is the same radical as in (1'). Eliminating the radical gives

PQ = 2 (R + N(N - 2)r) / (N - 1),

which, for N = 3, gives the required result.

Now, let's denote ρ = Nr, which is the radius of the circle (the dashed circle that appears when the box "extra" is checked) that has the small circles lined up on its diameter. Then we have

(3)

PQ = 2 (R + (N - 2)ρ) / (N - 1),

or, for N = 3, PQ = R + ρ. It's a pity we need a more complicated formula (3) in the general case.

A very enticing observation has been made in an article by J. Marshall Unger (Problem #5). Imagine the dynamics of moving from circle O'(ρ) to the circle O(R) while the line PQ and the small circles remain untouched. P'Q' is the corresponding chord in O'(ρ). Let a and b be the distances between the top and, respectively, bottom points of circles O'(ρ) and O(R) so that the change of radius is (b - a)/2:R - ρ = (a - b)/2. Then the change in the chord length is PQ - P'Q' = (b - a)/N. (I assume b > a). Moreover, if we introduce ρN = ρ - a/N and Nρ = ρ + b/N then, as illustrated in the applet, the two circles O'(ρN) and O'(Nρ) - the dotted circles in the applet - meet O(R) at exactly the points Q and P!